The most rigorous way to introduce relativity in the modeling of molecular systems is to use the four-component formalism derived from the Dirac equation. The method of choice is density functional theory (DFT) if many electrons are involved, as is the case with large metal clusters. In DFT, which is normally cast in the form of the independent-particle Kohn-Sham model, all of the exchangecorrelation effects are expressed implicitly as a functional of the electron density or, more generally, of the charge current density. The relativistic four-component generalization of the Kohn-Sham method, usually referred to as the Dirac-Kohn-Sham (DKS) model, was introduced several years ago. BERTHA is a modern implementations of the full four-component (4c) DKS formalism that is particularly appealing because it affords great physical clarity and represents the most rigorous way of treating explicitly and ab initio all interactions involving spin, which are today of great technological importance. However the full 4c DKS calculations have an intrinsically higher computational cost. However we have shown in a series of works that you can greatly reduce the weight of the computational burden of a DKS calculation, by implementing various parallelization and memory distribution schemes (L. Storchi et al. J. Chem. Theory Comput., 6, 384, 2010, J. Chem. Theory Comput., 9, 5356, 2013, S. Rampino et al. J. Chem. Theory Comput., 10, 3766, 2014) and by introducing new algorithms, such as those based on the method "density fitting" (L. Belpassi et al. J. Phys. Chem., 2006, 124, 124104; Phys. Rev. B, 2008, 77, 233403; J. Phys. Chem., 2008, 128, 124,108). This makes possible to carry out DFT calculations at full relativistic 4c level in an extremely efficient way. New perspectives of development are possible thanks to the above algorithmic advances which have represented a leap forward of several order of magnitude in the performance of the full DKS approach. These implementations have been carried out in an effective 4c code: BERTHA. The BERTHA code is basically built around a smart and efficient algorithm for the analytical evaluation of relativistic electronic repulsion integrals, developed by Quiney and Grant in Oxford more than a decade ago (I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theo. and Comp., Springer, 2007), which represents the relativistic generalization of the well-known McMurchie-Davidson algorithm. The basis functions employed are termed G-spinors, and are two-component spin-orbit-coupled functions derived from spherical Gaussian type functions. Up to that time 4c calculations had generally been applied to diatomic or polyatomic molecules with at most one heavy element. The current implementation of BERTHA, exploiting density fitting techniques and parallelization strategies, has extended the applicability range of all-electron DKS calculations to large clusters of heavy metals up to Au32. As mentioned above, these achievements represent the state of the art for full 4c calculations.
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